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X(2), X(13), X(14), X(523)

Ha, Hb, Hc projections of G on the altitudes

centers of the 6 equilateral triangles erected on the sides of ABC, externally (Ae, Be, Ce) or internally (Ai, Bi, Ci)

other points described below

K733 is a circular cubic with singular focus the reflection F of X(110) about X(2). SEARCH = 4.35117461087575. Note that F lies on the real asymptote of the Neuberg cubic K001. F is now X(9140) in ETC.

K733 meets the Napoleon cubic K005 at 9 identified points namely Ha, Hb, Hc, Ae, Be, Ce, Ai, Bi, Ci.

It also contains :

• A', B', C' on the perpendicular bisector (L) of X(13), X(14) and on the sidelines of triangles AeBeCe, AiBiCi. Recall that these two latter triangles are perspective at X(3) with perspectrix (L) hence A' = BeCe /\ BiCi.

• A", B", C" intersections of the parallels at X(2) to the sidelines of ABC with the cevian lines of X(523) i.e. A" = X(2)Ha /\ X(523)A.

• Y = X(14995), the third point on the Fermat axis and on the perpendicular at X(2) to the Euler line.

The real asymptote of K733 is the perpendicular at X(110) to the Euler line meeting K733 again at X = X(14932) on the lines X(2)X(690), X(111)X(1640).

One remarkable thing to observe is that the polar conic (P) of X(523) in K733 is also the polar conic of X(30) in K001 and the polar conic of X(5) in K005. This is the diagonal rectangular hyperbola passing through X(1), X(5), X(30), X(395), X(396), X(523), X(1749) and the excenters. More generally, (P) is the polar conic of P in any pK(X6, P) with pivot P on (P). K129a and K129b are two other examples of such cubics.

K733 is a Psi-cubic as in Table 60.

K733 is also the isogonal circular pK with pivot X(523) in the triangle T with vertices X(2), X(13), X(14).

The isogonal transform of the line at infinity is the circumcircle HP of T called the Hutson-Parry circle with center X(8371).

The points F and X are antipodes on this circle.

The isogonal transform of (L) is the Kiepert hyperbola (of ABC).

It follows that the in/excenters of T and the isogonal conjugates of the 9 identified points above also lie on K733.

These in/excenters are the intersections of (P) and the bisectors at G in T which are the lines passing through G and X(1340), X(1341).

Note that {A', A"}, {B', B"}, {C', C"}, {X(2), Y} are pairs of isogonal conjugates wrt T.

Remark :

• The locus of points having the same isogonal conjugate in both triangles ABC and T is K018.

• the centroid of T is :

X(9166) = -a^4+a^2 b^2-4 b^4+a^2 c^2+7 b^2 c^2-4 c^4 : : , with SEARCH = 2.35958190543559.

• the orthocenter of T is :

X(9180) = (b-c) (b+c) / (2 a^4-2 a^2 b^2-b^4-2 a^2 c^2+4 b^2 c^2-c^4) : : , with SEARCH = 5.84101666957244.

• the Lemoine point of T is :

X(41939) = 2 a^6-2 a^4 b^2+b^6-2 a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-b^2 c^4+c^6 : : , with SEARCH = 1.95615000384013.

• the nine-point center of T is :

X(9183) = (b^2-c^2) (a^8-2 a^6 b^2+12 a^4 b^4-11 a^2 b^6+b^8-2 a^6 c^2-18 a^4 b^2 c^2+9 a^2 b^4 c^2+7 b^6 c^2+12 a^4 c^4+9 a^2 b^2 c^4-15 b^4 c^4-11 a^2 c^6+7 b^2 c^6+c^8) : : , with SEARCH = 3.22994059646980.